One easy recommendation is Shambles Square. There are two big pubs there and it sits in a square next to the Arndale and close to the Corner Exchange where there are places to eat, Urbis (Football Museum), Printworks (which houses some bars), and is near Manchester Cathedral, the Arndale and St Annes Sq (Royal Exchange Theatre is worth a look around if you have time).
Northern Quarter has a lot of bars too and a nice atmosphere including Afflecks Palace which certainly used to be worth walking around. Trendy place for young people although that was a long time ago when I was young!
General guide:
https://www.visitmanchester.com/
There are lots of parts of Manchester worth a visit but some bits are a little inconvenient if you are going to the football.
If your kids are thinking about uni - Manchester Museum is worth a visit as it's based in the Manchester University campus. And you can check out the campus which is really cool. The students aren't there now. But it's on a fantastic road for buses which takes you right into the heart of Manchester: Albert Sq and St Peters Sq worth a look. Depends how much time you have.